On the efficiency of some p-groups

Let p be a prime number. In this paper, we work on the efficiency of the p-groups G₁ and G₂ defined by the presentations,

here α ≥ β > γ ≥ 1 and

where α ≥ 2γ, β > γ ≥ 1 and α + β > 3. For example, if we let p = 2, then by [1], the groups defined by these presentations becomes 2-groups. It is known that these groups play an important role in the theory of groups of nilpotency class 2.